Examples

1.

If 𝔮 is reductive then ind⁡𝔮=rank⁡𝔮. Indeed, 𝔮 and 𝔮* are isomorphic as
representations for 𝔮 and so the index is the minimal dimension among stabilizers of elements in 𝔮. In particular the minimum is realized in the stabilizer of any regular element of 𝔮. These elemtents have stabilizer dimension equal to the rank of 𝔮.

2.

If ind⁡𝔮=0 then 𝔮 is called a
Frobenius Lie algebra. This is equivalent to condition that
the Kirillov formKξ:𝔮×𝔮→𝕂 given by (X,Y)↦ξ⁢([X,Y]) is non-singular for some ξ∈𝔮*. Another equivalent condition when 𝔮 is the Lie algebra of an algebraic group Q is that 𝔮 is Frobenius if and only if Q has an open orbit on 𝔮*.